Convex Order and Arbitrage

Wiesel and Zhang [2023] established that two probability measures $\mu ,\nu$ on ${\mathbb{R}}^d$ with finite second moments are in convex order (i.e. $\mu {⪯}_c\nu$) if and only if $W_2(\nu ,\rho {)}^2-W_2(\mu ,\rho {)}^2\le \int \mathrm{\mid }y{\mathrm{\mid }}^2\nu (dy)-\int \mathrm{\mid }x{\mathrm{\mid }}^2\mu (dx)\mathrm{.}$ Let us call a measure $\rho$ maximizing $W_2(\nu ,\rho {)}^2-W_2(\mu ,\rho {)}^2$ the optimal $\rho$. This paper summarizes key findings by Wiesel and Zhang, develops new algorithms enhancing the search of optimal $\rho$, and builds on the paper through constructing a model-independent arbitrage strategy and developing associated numerical methods via the convex function recovered from the optimal $\rho$ through Brenier’s theorem. In addition to examining the link between convex order and arbitrage through the lens of optimal transport, the paper also gives a brief survey of functionally generated portfolio in stochastic portfolio theory and offers a conjecture of the link between convex order and arbitrage between two functionally generated portfolios.